Anime? Rei and Shinji are the quintessential fighting duet... but my favorite might have to be Akane and Ranma, I love when they fight, even though it's mostly verbal. As they say, the pen is mightier than the sword.

Before they became the coolest pair of Samurai to ever wield a sword, there was this one of the ten best anime sword fights. It’s a brilliant display of their individual skills and lays the ground work for their teaming up.

Fairy Tail
Happy & Lucy: Happy makes a joke or comment, and Lucy over reacts to it. Sometimes threatening Happy. When Natsu and Happy do something weird at Lucy's expense, it's always good. Lucy and her summons are runner ups.

SKET Dance
Bossun & Tsubikari: Everytime these two end up with some kind of shared task or competition, it's always entertaining.

Asura's Wrath
Asura & Yasha: Epic fights between these two. They're like brothers, though they pretend to hate each other. Intense rivalry. Sometimes working together, other times fighting each other till they both have little or no strength left. Runner up, is Augus, the mentor of Asura and Yasha. Play the game, you'll understand completely. This game, is practically an interactive anime itself.